Phase-stepping interferometry

ABSTRACT

An interferometric procedure, such as electronic speckle pattern interferometry, involves generating two signals representing the point-by-point variations in intensity of respective patterns of electromagnetic radiation resulting from the interference of first and second beams of such radiation derived from a coherent source, with at least the first beam from each pattern being scattered, before interference with its respective second beam, from a common object surface, and with a corresponding pair of the beams, one for each pattern, having a predetermined relative phase difference of other than a multiple of π; and determining from the two signals values for a datum phase of the radiation at the object surface. Preferably, as a preliminary to this last determination. DC components are removed from the two signals. Conveniently, to simplify the determination, the phase difference is an odd multiple of π/4 or π/2.

The present invention has been conceived and developed for extractingphase information in electronic speckle pattern interferometry, orso-called ESPI, by the phase-stepping technique and it is convenient todescribe the invention in this context. However, it is to be understoodthat the invention is applicable to other forms of interferometry forwhich the phase-stepping technique is appropriate.

This technique involves successively changing the phase of one of theinterferometer beams from a datum phase φ by 2π/k in (k-1) equal stepsand recording the respective interference pattern images generated ateach different phase. The Fourier series coefficients for the datumphase are then given by the expressions ##EQU1## where N is the imagenumber in the phase-stepping sequence and I_(N) (x,y) is the image pixelintensity. A solution for φ(x,y) is then given, in turn, by theexpression

    tan.sup.-1 (β.sub.1 (x,y)/α.sub.1 (x,y))

However, the technique is not without practical difficulties.

For a given image resolution, image recording and related equipmentneeds increase with k and so also, in general terms, will the time takenfor computation by virtue of the increased number of pieces of datainvolved in the expressions to be evaluated.

At the same time, the environmental sensitivity of the opticalinstrumentation can drift and introduce error in I_(N) as k increases.

In practice these difficulties are such that the technique is usuallydeployed with k equal to 3 or 4, when evaluation is respectively givenby ##EQU2## When k is 3 the expression to be evaluated iscomputationally more complex than that for k equal to 4, butenvironmental sensitivity is higher. Higher values of k are relativelyimpracticable by virtue of greatly increased equipment cost and/orcomplexity plus poor sensitivity.

Clearly this situation is such that the case when k is 2 appearsparticularly attractive. However, in this case it will be seen that β₁is zero.

An object of the present invention is to reduce the difficulties of theabove situation and to this end there is provided an interferometricmethod comprising, and related apparatus for, generating two signalsrepresenting the point-by-point variations in intensity of respectivepatterns of electromagnetic radiation resulting from the interference offirst and second beams of such radiation derived from a coherent source,with at least the first beam for each pattern being scattered, beforeinterference with its respective second beam, from a common objectsurface, and with a corresponding pair of the beams, one for eachpattern, having a predetermined relative phase difference of other thana multiple of π; and determining from said two signals values for adatum phase of the radiation at said surface.

The derivation of the invention can be explained by considering, forESPI, the case for k equal to 3 to produce images at phases φ-δ, φ andφ+δ. These images can be expressed in a simplified manner by theequations ##EQU3## Included in these equations are DC components I_(O)plus I_(R) contributed by the object and reference beams producing theimages. Elimination of these components, such as by selective frequencyfiltering or substraction, allows further simplification effectively torepresent a situation where k is 2 by forming from only two of theequations an expression for φ.

Thus from equations (1) and (2), φ is given by ##EQU4## and if δ ischosen for convenience as π/2 or an odd multiple thereof, thisexpression is further simplified to

    tan.sup.-1 ±I.sub.1 /I.sub.2

A corresponding result is obtainable from equations (2) and (3).

From equations (1) and (3), φ is given by ##EQU5## and if δ is chosenfor convenience as π/4, or an odd multiple thereof, cot δ equals ±1.

For further clarification the accompanying drawing schematicallyillustrates, by way of example, one embodiment of electronic specklepattern interferometric apparatus according to the invention.

The illustrated apparatus comprises a coherent light source, laser 1,which projects a beam by way of a beam splitter 2 to an object surface 3of interest and a reference surface 4, with a television camera 5 orequivalent equipment serving to image light scattered from both surfacesand provide a video signal output representing the relevant image. Aphase shifter 6 is located in the beam path from the beam splitter toone of the surfaces 3 and 4, this shifter being of electro-optical orother form selectively operable to effect a predetermined phase shiftwithin the relevant beam portion. The video signal is applied through afilter 7, operable to remove DC components, and then a switch 8,operable in synchronism with the phase shifter to pass alternate ones ofa succession of video signals to a frame store 9. A successive pair ofvideo signals are passed, one from the store and one directly by theswitch, to a processor 10, with the processor output being passed to anoutput unit 11 for the purposes of recordal and/or display.

The phase shifter will be operable to effect a relative phase shift of δor 2δ, video signals representing images formed respectively with andwithout this shift will be passed as parallel inputs to the processor bydifferent ones of the routes through the frame store or not, and theprocessor will determine pixel-by-pixel from these inputs, output valuesfor a datum phase φ on the basis of the earlier discussion of theinvention above. Thus, if the relative phase shift is δ thedetermination follows that for equations (1) and (2) or (2) and (3),while if the shift is 2δ the determination follows that for equations(1) and (3).

It will be noted that phase shifting can be effected in respect of thebeam portion incident on the object surface or reference surface, butnot both. Also the beam can be generated in continuous or pulsed mannerwith, in the latter case, phase shifting and frame store input switchingbeing suitably synchronised with the pulse timing. Furthermore, to theextent that the datum phase determination is effectively absolute, thereneed be no change in condition for the object surface, such as caused bydeformation or other movement, as is commonly the case for electronicspeckle pattern interferometry, although the invention is of courseapplicable to such situations.

From this clarification it will be seen that the invention provides abasis for improved phase-stepping interferometry or apparatus thereforinvolving a single phase change step, simplified equipment andcomputation, and enhanced sensitivity. Moreover, as noted initiallyabove, while conceived and developed for electronic speckle patterninterferometry, the invention is more generally applicable tointerferometric procedures and apparatus involving coherent light orother electromagnetic radiation and for which the phase-steppingtechnique is appropriate, such as holographic interferometry.

We claim:
 1. An interferometric method comprising: generating twosignals representing point-by-point variations in intensity ofrespective patterns of electromagnetic radiation resulting from aninterference of first and second beams of such radiation derived from acoherent source, with at least the first beam for each pattern beingscattered, before interference with its respective second beam, from acommon object surface, and with a corresponding pair of the beams, onefor each pattern, having a predetermined relative phase difference ofother than an integer multiple of π; anddetermining, from said twosignals alone, values for a datum phase of the radiation at saidsurface.
 2. A method according to claim 1 wherein, as a preliminary tosaid datum phase value determination, DC components are removed fromsaid two signals.
 3. A method according to claim 1 or 2 wherein saidphase difference is an odd multiple of π/2.
 4. A method according toclaim 3 wherein each of said corresponding pair of beams has a phasedifference which is the same odd multiple of π relative to said datumphase but of mutually opposite sense.
 5. A method as in claim 3, whereinsaid determining step includes the steps of:obtaining equationsrepresenting said two signals, such that ##EQU6## such that when δ is anodd multiple of π/2,

    φ=tan.sup.-1 ±I.sub.1 /I.sub.2 ; and

solving said equations for said datum phase.
 6. A method as in claim 1,wherein said determining step includes the steps of:obtaining equationsrepresenting said two signals, in a way that a phase shift amountbecomes ±1; and solving said equations for said datum phase.
 7. A methodaccording to claim 1 wherein said radiation is light.
 8. Interferometricapparatus comprising:means for generating a first signal representingpoint-by-point variations in intensity I of a pattern of electromagneticradiation resulting from an interference of first and second beams ofsuch radiation derived from a coherent source, with at least the firstbeam being scattered from an object surface before interference with thesecond beam; means for changing the phase of one of said first andsecond beams by an amount δ other than an integer multiple of π togenerate a second signal in like manner to said first signal; means forremoving DC components from said first and second signals; and means,responsive only to said first and second signals following DC componentremoval, to evaluate a datum phase φ by solving two equations of threein the form I=cos (φ-δ), cos φ and cos (φ+δ).
 9. Apparatus according toclaim 8 comprising means for storing at least one of said first andsecond signals.
 10. Apparatus according to claim 8 or 9 wherein saidsource is of laser form.
 11. An apparatus as in claim 8, wherein saidevaluating means evaluates equations representing said two signals,including ##EQU7## such that when δ is an odd multiple of π/2, φ=tan⁻¹±I₁ /I₂ ; and solves said equation for said datum phase.